Work

Solving the Fractional Allen–Cahn Equation and the Fractional Cahn–Hilliard Equation with the Fractional Physics-Informed Neural Networks

Year: 2025

Type: article

Abstract: In this paper, we present a computational framework based on fractional Physics-Informed Neural Networks combined with the L1 approximation of the Caputo fractional derivative to solve the fractional Allen–Cahn and fractional Cahn–Hilliard equations. Considering the limitation of the original fractional Physics-Informed Neural Networks in achieving high accuracy when applied to these highly stiff fractional equations, we propose three improved optimization strategies: adaptive non-uniform sampling, adaptive exponential moving average ratio loss weighting, and two–stage adaptive quasi optimization. By combining these strategies, three improved fPINNs algorithms are developed: f–A–PINNs, f–A–A–PINNs, and f–A–T–PINNs. Numerical experiments demonstrate that the f–A–T–PINNs algorithm achieves superior computational accuracy and improved parameter stability compared to the other algorithms.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract9120773

Authors: Xiaorong Kang, Yang Li, Yongzheng Li, Jinsong Hu, Kelong Zheng

Institutions: Southwest University of Science and Technology, Civil Aviation Flight University of China, Chengdu Technological University

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258

Funder: Fundamental Research Funds for the Central Universities